(a) Calculate the vertical velocity V0 achieved after the first twenty seconds of vertical flight from the lunar surface. Estimate the altitude gain. (b) After twenty seconds of ve... (a) Calculate the vertical velocity V0 achieved after the first twenty seconds of vertical flight from the lunar surface. Estimate the altitude gain. (b) After twenty seconds of vertical flight, the vehicle is rapidly pitched through twenty degrees, and its new attitude maintained constant thereafter by an automatic control system. Estimate the vehicle's velocity vector and speed after a further one hundred and twenty seconds of powered flight. Read more

Steps to Solve

1. Calculate the altitude gain after 20 seconds of vertical flight

Assuming constant acceleration due to gravity on the Moon, which is approximately $1.625 , \text{m/s}^2$, we can find the altitude gain using the formula for distance under constant acceleration:

$$ d = v_0 t + \frac{1}{2} a t^2 $$

Assuming the initial velocity ($v_0$) is 0, and $t = 20 , \text{s}$:

$$ d = 0 + \frac{1}{2} (1.625) (20^2) $$

Calculating gives:

$$ d = \frac{1.625}{2} \times 400 = 325 , \text{m} $$

So, the altitude gain after 20 seconds is 325 meters.

2. Calculate the vertical velocity after 20 seconds of flight

The vertical velocity can be calculated using the formula:

$$ v = v_0 + a t $$

Again, assuming $v_0 = 0$ and using $a = 1.625 , \text{m/s}^2$ and $t = 20 , \text{s}$:

$$ v = 0 + (1.625)(20) $$

Calculating gives:

$$ v = 32.5 , \text{m/s} $$

So, the vertical velocity after 20 seconds is 32.5 m/s.

3. Estimate the vehicle's velocity vector after 140 seconds of powered flight at 20 degrees

To find the velocity vector after an additional 120 seconds (total of 140 seconds) with a directed thrust at 20 degrees, calculate the vertical and horizontal components.

Assuming a constant thrust acceleration, the velocity after 140 seconds can be calculated similarly:

$$ v = v_0 + a t $$

However, we have to separate this into vertical ($v_y$) and horizontal ($v_x$) components.

Calculating the total vertical velocity contribution after 140 seconds:

$$ v_y = (1.625)(140) $$

Calculating gives:

$$ v_y = 227.5 , \text{m/s} $$

To find the horizontal component based on the pitch angle:

$$ v_x = v \cos(20^\circ) $$

Now we need to apply cosine:

$$ v_x = 227.5 \cos(20^\circ) $$

Calculating gives approximately:

$$ v_x \approx 213.1 , \text{m/s} $$

4. Calculate the magnitude of the total velocity vector

The total velocity can now be calculated using the Pythagorean theorem:

$$ v_{total} = \sqrt{v_x^2 + v_y^2} $$

Substituting the values we found:

$$ v_{total} = \sqrt{(213.1)^2 + (227.5)^2} $$

Calculating gives:

$$ v_{total} \approx \sqrt{45324.61 + 51756.25} = \sqrt{97080.86} \approx 311.5 , \text{m/s} $$